Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle MON = 4x - 34$, and $ m \angle LOM = 7x - 63$, find $m\angle MON$. $O$ $L$ $N$ $M$
Explanation: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {7x - 63} + {4x - 34} = {90}$ Combine like terms: $ 11x - 97 = 90$ Add $97$ to both sides: $ 11x = 187$ Divide both sides by $11$ to find $x$ $ x = 17$ Substitute $17$ for $x$ in the expression that was given for $m\angle MON$ $ m\angle MON = 4({17}) - 34$ Simplify: $ {m\angle MON = 68 - 34}$ So ${m\angle MON = 34}$.